Bottleneck Profiles and Discrete Prokhorov Metrics for Persistence Diagrams

Abstract

In topological data analysis (TDA), persistence diagrams have been a succesful tool. To compare them, Wasserstein and Bottleneck distances are commonly used. We address the shortcomings of these metrics and show a way to investigate them in a systematic way by introducing bottleneck profiles. This leads to a notion of discrete Prokhorov metrics for persistence diagrams as a generalization of the Bottleneck distance. They satisfy a stability result and bounds with respect to Wasserstein metrics. We provide algorithms to compute the newly introduced quantities and end with an discussion about experiments.

34 pages, 12 figures; improved exposition. To appear in Discrete & Computational Geometry

References in corpus (4)

• Scikit-learn: Machine Learning in Python
• On the stability of persistent entropy and new summary functions for TDA
• Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport
• Topological spaces of persistence modules and their properties